Answer
$x^{2} +3x +1$
Work Step by Step
$\begin{array}{ccccccccccc}
& &x^{2} & +3x & +1 & \\
& &--&--&--&--\\
x+2&) & x^{3} &+5x^{2} &+7x& +2 & \\
& & x^{3}&+2x^{2} & & & \color{red}{\leftarrow \small{x^{2}(x+2) } } \\
& &--&-- & & &\color{red}{ \small{subtract}} \\
& & & 3x^{2} & +7x& +2 & \\
& & & 3x^{2} & +6x & & \color{red}{\leftarrow \small{3x(x+2) } }\\
& & &--&-- & &\color{red}{ \small{subtract}} \\
& & & & x & +2 &\\
& & & & x & +2 & \color{red}{\leftarrow \small{1(x+2) } }\\
& & & &-- &-- & \color{red}{ \small{subtract}} \\
& & & & & 0
\end{array}$
Quotient = $x^{2} +3x +1$
Remainder = $0$.
$\displaystyle \frac{dividend}{divisor}=quotient+\frac{remainder}{divisor}$
$\displaystyle \frac{ x^{3} +5x^{2} +7x +2}{x+2}$ = $x^{2} +3x +1$
